Table of Contents

The JPL solar system ephemeris has a wide variety of applications for
spacecraft navigation and ground-based astronomy. I shall describe only
two applications of relevance to the GBT, generation of lunar and planetary
positions for telescope tracking and pulsar synchronization and data
reduction.

The ephemeris contains equation coefficients which describe the
three-dimensional positions and velocities of the sun, moon, and planets as
a function of time in a rectangular coordinate system. There is a set of
coefficients for each coordinate component of position and velocity for
each object. Each set of coefficients typically covers a time span of
between 8 and 32 days. The number of coefficients and time span depends on
the rate of change and required accuracy of an object's position and
velocity. See the documents on the descriptions of the JPL Ephemeris, and the computer code for reading the ephemeris for
more details.

The X-Y plane of the ephemeris rectangular coordinate system is parallel
to the earth's equatorial plane. The X axis is in the direction of the
vernal equinox, the Y axis points toward RA = 6 hours, Dec = 0, and the Z
axis is in the direction of the north celestial pole. The position
components are in kilometers, and the velocity components are in kilometers
per day. The time independent variable for the equations is Terrestrial Dynamic Time in fractional Julian
days.

The origin of the rectangular coordinates is at the solar system
barycenter (center of mass) for the sun and all planets, except the earth.
The solar system barycentric positions of the earth and moon may be derived
from the ephemeris specifications of the position and velocity
of the earth-moon barycenter and the geocentric position and velocity of
the moon.

Since the orientation of the earth's equator and pole are is
continuously changing, the coordinate system must be specified for a
particular epoch. The DE100-series ephemerides use the B1950 equator and
equinox. The DE200 series uses the J2000 system [ref
1]. The most recent DE400 series are also in the J2000 system, but its
equator and equinox are defined by the reference frame of the International
Earth Rotation Service (IERS), which is more
precisely tied to distant celestial objects than is the IAU J2000 standard.

There is an interesting change in computational perspective brought about
by the three dimensional nature of the JPL ephemeris. As classical
astronomers we are most familiar with the two dimensional world of
spherical direction coordinates. Adding the third dimension to spherical
coordinates for nearby objects is much more complicated than making a
complete switch to rectangular coordinates for all calculations except the
final conversion to spherical coordinates for the purposes of pointing a
telescope. Even precession, nutation, and aberration (the tedium of
every observer's life) are more easily visualized and specified for the
computer in three-dimensional coordinates.

The solar system is not static. Apparent directions are
significantly affected by the light travel time between two objects
as explained below.

Earthly observers are interested in the direction of the moon and planets
as seen from our observatories. The ephemeris can give us only geocentric
positions, however. For the moment we shall assume that we have a
computational source for the observatory's
geocentric position and proceed with a brief description of computing
the apparent position of an object in the sky.

To get the instantaneous topocentric (observatory's) position of the
moon we add vectors from the observatory to the earth's center, and from
the earth's center to the moon's center. The latter is supplied by the
ephemeris. If we let (x,y,z) be geocentric and (a,b,c)
be topocentric vector coordinates, the vector equation is

Moon(a,b,c) = Moon(x,y,z) - Observatory(x,y,z)

Using available ephemeris information, a planet's topocentric position is
the vector sum: observatory-&gt(earth's center) +(earth's
center)-&gt(earth-moon barycenter) + (earth-moon barycenter)-&gt(solar
system barycenter) +(solar system barycenter)-&gtplanet.

If we let (X,Y,Z) be solar system barycentric coordinates,
EMB represent the earth-moon barycenter, and EMrat be the
earth/moon mass ratio, then

Specifying a planet's position in the sky can be quite confusing because it
can be done in any one of a number of spherical coordinate
conventions (B1950, J2000, current epoch, etc.), and there are two physical
corrections for the finite speed of light. We will say more about the coordinate conventions in a separate document.
For the moment let's look at the light travel time corrections

Things are always as they appear, but it depends on whom you ask. When
we look at a planet we are actually seeing the planet where it was when its
light left the planet. This could be minutes or even hours before the
current time. The procedure for compensating for this time delay is to
compute the distance to the planet at the time of observation. From
this compute the light travel time, recompute the planet's position for
current time minus light travel time, and use this earlier planet's
position with the current observatory, moon, and earth-moon
barycenter positions in the Planet(a,b,c) equation above.

The planet's position obtained from this procedure
is its "astrometric" position in the reference frame of the solar system
ephemeris. That is, it is the position of the planet as it would appear on
the background of stars as plotted in this frame, J2000 for
example.

The second correction to the apparent direction of a planet, due to the
finite speed of light, comes from the motion of the observer. The same
correction needs to made to star positions, where it is called "aberration
of star light." The time-worn analogy is of a person running in the rain.
If the person is standing still, the rain appears to be coming straight
down, but, if the person is moving, the rain appears to be coming from the
direction of motion. The aberration correction to the apparent direction
of a star or planet, in radians, is the ratio of the velocity component of
the observer's motion perpendicular to the line of sight to the speed of
light. The earth's orbital velocity is about 30 km/s so the annual
aberration can be much as 30 / 300,000 = .0001 radians or about 20
arcseconds.

One might ask "velocity with respect to what?" when computing
aberration. In the case of stars we dodge the question by computing only
differential aberration for different times of day and year,
hence, the terms "diurnal aberration" and "annual aberration." For a
planet we can use the observer's lateral velocity with respect to the
planet, but this correction will include the time-of-flight correction for
the speed of light outlined above. Since the time-of-flight correction
used the planet's velocity with respect to the solar system barycenter, we
can add the aberration correction using the earth's velocity with respect
to the barycenter. Another, slightly more rigorous approach is to compute
the sum of both corrections together by computing the direction of the
planet using the positions of both the planet and observer at the current
time minus the light travel time.

The Astronomical Almanac is not entirely consistent on how it lists
planetary positions. Pluto's position is tabulated as astrometric. In
other words, it is corrected for the light's time-of-flight but not for
aberration, and the coordinate frame is J2000 as given by the DE200
ephemeris. All other planetary positions are listed as "apparent", the
position with respect to the current equator and equinox including the
correction for aberration. In other words, these positions are corrected
for precession, nutation, aberration, and time-of-flight.

The choice of coordinate system for an ephemeris is an arbitrary one. In
the past the strongest criterion was that the coordinate system be one in
very common use. The advent of much more accurate celestial measurement
techniques and fast computers for coordinate conversions has changed the
dominant criterion to the precision with which a coordinate system can be
tied to very distant celestial objects. The JPL ephemerides have followed
this evolution with the DE100, DE200, and DE400 series, which are based on
the B1950, J2000, and IERS coordinate systems, respectively.

The B1950 and J2000 equatorial coordinate systems are defined by the
mean orientation of the earth's equator and ecliptic at the beginning of
the years 1950 and 2000. The assumed orientation of the earth on these two
dates is more a matter of definition than actual, since the 'mean'
orientation does not include short term motions of the earth's spin axis
(nutation and smaller effects). Of course, he coordinates of celestial
objects in B1950 and J2000 differ by many arcminutes.

The B1950 system is tied to the sky by star coordinates in the FK4
catalog, and J2000 is tied to FK5. Because star catalogs tend to be
weighted toward nearby stars, they are subject to stellar proper motions
and the assumption that the motions of many stars will average out. Since
the FK4 catalog was published, offsets and drift rates have been determined
for its average stellar positions at the level of a few hundredths of an
arcsecond as summarized in [ref 2]. These corrections
are insignificant for telescope pointing, but they are important for VLBI
and pulsar observations.

The IERS
reference frame is essentially the J2000 system except that it is tied to the
sky by the published positions of 228 radio sources, roughly 23 of which
are monitored by several VLBI networks to determine the day to day changes
in the orientation of the earth [ref 3]. These radio
sources are very distant compared to stars and should not suffer any of
the proper motion problems. The DE403 and IERS celestial frames are tied
together to an accuracy of a few milliarcseconds [ref
5]. The IERS and FK5 systems are consistent to the accuracy of the FK5
catalog.

The JPL ephemerides themselves define a celestial reference frame since
they publish the positions of solar system objects in space in a coordinate
system which is presumably fixed with respect to distant celestial objects.
Although every effort has been made to tie the ephemeris dynamical frame to
a stellar frame, there will always be an offset at some level. You
must be aware that, when working with more than one ephemeris or coordinate
system, you need to take into account the accuracy with which these systems
are known to be tied. Avoid switching systems unnecessarily, and define
your assumptions and choices clearly for other observers using your data.

Prediction and reduction of pulsar pulse times of arrival make use of the
ephemeris information on the earth's changing distance to a pulsar as it
moves around the sun. Garden variety pulsar timing can achieve accuracies
of tens of microseconds, and millisecond pulsar timing is working at the
level of a few hundred nanoseconds over long periods of time. The latter
number implies that the observer's position with respect to the solar
system barycenter be known to about 100 meters. This is very close to the
accuracy limit of the latest ephemerides.

Pulsar timing also requires a relativistic gravitational correction due
to the earth's motion in the solar system's potential well. This
correction can be as large as 1.6 milliseconds. The standard pulsar
timing reference is the solar system barycenter using Barycentric Dynamic Time.

The geometric timing correction is simply the speed of light divided
into the distance from the observer to the solar system barycenter times
the cosine of the angle between the observer-barycenter line and the
direction to the pulsar. In vector language this is the dot product of the
observer's barycentric vector and the pulsar direction unit vector divided
by the speed of light. Using the ephemeris vector notation above, the
observer's barycentric vector is

The main caution is to keep the pulsar's position in the same coordinate
frame as the ephemeris. A 0.1 arcsecond pulsar direction error can produce
a timing error as high as 240 microseconds, so all of the reference frame
caveats of the previous section apply here, too. Uncertainties involved
with converting from B1950 to J2000 coordinates, for instance, can have
effects at the 10 to 100 microsecond level.

Equations for the relativistic time correction are given in the
Explanatory Supplement to the Astronomical Almanac [ref
4].